Definition 1

— A subset $A$ of an affine space $E$ is convex if, for every two points $x, y$ of $A$, the closed segment with end points $x, y$ is contained in $A$.

As $(1 - \lambda) a + \lambda x = a + \lambda (x - a)$, this definition is equivalent to the following: the set $A$ is convex if, for every point $a \in A$, the transform of $A$ by a homothety of centre $a$ and ratio $\lambda$ where $0 < \lambda < 1$, is contained in $A$ (in other words, $A$ is stable for these homotheties).